Integral of dx/x(x^2+1) \[ \int\:\:\frac{\boldsymbol{{dx}}}{\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}} \:+\:\mathrm{1}\right)} \]
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\[\:\:\:\:
\int\:\frac{\boldsymbol{{dx}}}{\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}}+\mathrm{1}\right)} \\
=\:\int\:\frac{\left(\boldsymbol{{x}}^{\mathrm{2}}+\mathrm{1}\right) −\boldsymbol{{x}}^{\mathrm{2}} }{\boldsymbol{{x}}\left(\boldsymbol{{x}}^{\mathrm{2}}+\mathrm{1}\right)}\boldsymbol{{dx}} \\
=\:\int\:\frac{\boldsymbol{{dx}}}{\boldsymbol{{x}}}−\int\frac{\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}\boldsymbol{{dx}} \\
=\:\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{e}}}\mid\boldsymbol{{x}}\mid−\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{2}\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}\boldsymbol{{dx}} \\
=\:\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{e}}}\mid\boldsymbol{{x}}\mid −\frac{\mathrm{1}}{\mathrm{2}}\int \frac{\frac{\boldsymbol{{d}}}{\boldsymbol{{dx}}}\left(\boldsymbol{{x}}^{\mathrm{2}}+\mathrm{1}\right)}{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}\boldsymbol{{dx}} \\
=\:\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{e}}}\mid\boldsymbol{{x}}\mid −\frac{\mathrm{1}}{\mathrm{2}} \int \frac{\boldsymbol{{d}}\left(\boldsymbol{{x}}^{\mathrm{2}} + \mathrm{1}\right)}{\boldsymbol{{x}}^{\mathrm{2}} + \mathrm{1}} \\
=\:\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{e}}}\mid\boldsymbol{{x}}\mid −\frac{\mathrm{1}}{\mathrm{2}} \boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{e}}} \mid\left(\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}\right)\mid \\
=\:\boldsymbol{\mathrm{log}}_{\boldsymbol{\mathrm{e}}}\mid\frac{\boldsymbol{{x}}}{\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\mathrm{1}}}\mid + \boldsymbol{{c}}
\]
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